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A team ships only as fast as its slowest part. Push hard while a teammate coasts and your effort is wasted; the project still moves at the laggard’s pace. So how hard should you push, when it depends on everyone else pushing too?

Supermodular games

In some games your incentive to do more grows the more everyone else does. These strategic complements have a striking feature: many coordinated outcomes are all equilibria, ranked from best to worst, and fear of the low one can trap everyone there even though all would prefer the high one.

From earlier lessons

Two quick ones before we start.

A finite game has no stable pure-strategy outcome. What can still exist?

A focal point helps players pick an equilibrium using...

Predict, then play

You and three teammates each pick an effort from 1 to 7. Your payoff is $0.6 + 0.2\times(\text{group minimum}) - 0.1\times(\text{your own effort})$ . The reward tracks the weakest link, so your effort helps only if it does not exceed everyone else’s.

Why did that happen?

Take a guess, then reveal the reason

Every unit of effort costs you, but the reward depends only on the smallest effort in the group. So raising yours above the minimum spends the cost and buys nothing. That makes your best move: match the lowest effort you expect. When you fear a teammate will play low, you play low too, and so does everyone, which is exactly how a group of people who would all prefer to work hard end up stuck working little. Note which way the ratchet turns: one low effort drags the group down at once, and no amount of effort from you alone lifts it back, because your extra work cannot raise anyone else’s minimum. Van Huyck’s groups slid to the floor over repeated rounds for exactly this reason.

The math

With payoff $u_i = 0.6 + 0.2\,\min_j e_j - 0.1\,e_i$ , consider everyone choosing the same effort $x$ . Then $\min_j e_j = x$ and each earns $0.6 + 0.1x$ , rising in $x$ . Every common effort from 1 to 7 is a Nash equilibrium, and they are Pareto-ranked: all-7 pays $1.3$ , all-1 pays $0.7$ .

Why are they all equilibria? Given the others at $x$ , raising your effort leaves the minimum at $x$ (so no gain) but costs more; lowering it drops the minimum and your reward. So matching is a best response.

Definition, supermodular game (strategic complements)

On ordered strategy sets, a game has strategic complements when each player’s payoff has increasing differences in their own action and the others’ actions, so higher actions by others make higher own actions more attractive. With compact complete-lattice strategy sets, suitable continuity, and these complementarity conditions, the equilibrium set has a smallest and a largest element. The finite minimum-effort game above satisfies those assumptions.

The danger is selection. All-7 is best for everyone, but it is risky: one cautious teammate and your high effort is wasted. All-1 is safe. Strategic uncertainty, not payoffs, is what pulls groups toward the low equilibrium.

Work it out

Why matching is the best response

Suppose you expect the others’ minimum to be $m$ . If you play $e > m$ , the group minimum is still $m$ , so your payoff is $0.6 + 0.2m - 0.1e$ , which falls as $e$ rises. If you play $e < m$ , the minimum drops to $e$ , giving $0.6 + 0.2e - 0.1e = 0.6 + 0.1e$ , which is lower than matching at $m$ . So the best response is $e = m$ : match the minimum you expect.

Because that best response increases with what others do, the equilibria line up from all-1 to all-7. All-7 gives everyone $1.3$ ; all-1 gives $0.7$ . Everyone prefers all-7, yet all-1 is perfectly stable, held in place by the fear that someone will slack.

Key idea

When efforts are complements, best responses move together, so coordinated outcomes stack up as equilibria. The best one is fragile: fear of a weak link can lock everyone into the worst.

Check yourself

Check yourselfapply

You expect your teammates' lowest effort to be 4. What is your best response?

Check yourselfanalyse

All-7 pays everyone 1.3 and all-1 pays 0.7. Both are equilibria. Why might a group end up at all-1?

Check yourselfbrings back: focal points

Both all-7 and all-1 are equilibria with identical structure. What could tip a group toward the good one?

Sources

Van Huyck, J. B., Battalio, R. C., and Beil, R. O. (1990). Tacit coordination games, strategic uncertainty, and coordination failure. American Economic Review, 80(1), 234-248.
Milgrom, P., and Roberts, J. (1990). Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58(6), 1255-1277.